I wonder if there's a way to tell without injuring the tree.
There sure is; use a tool called an increment borer. You screw a long tube into the tree and with an extractor pull out a small diameter coring. The rings can then be counted. The core can be saved or put back into the tree, hole sealed up and the tree is none the worse for the experience. You can gather a lot of information on the tree's growth rate and infer climate or weather conditions during the tree's life to that point. Since I doubt a borer is made long enough to reach the center of the tree, you'd take a sample boring, up to several feet long, measure growth rate and infer the age.